where \(\mbox{$B$}{}^e_r\hat{\mbox{$u$}{}}^e_r\)  is the approximation of the gradient as shown in equation REF  for a particular triangle \(r\) . Here, the vector \(\hat{\mbox{$u$}{}}^e_r\)  is the prediction of the neural network at the nodes of the triangle \(r\) , which depend on the trainable parameters \(\mbox{$\theta $}{}\) . We use 50 eigenfuctions of the Laplace-Beltrami operator as input of the neural network, which is then followed by a single hidden layer with 100 neurons and hyperbolic tangent activations. We train the neural network with the ADAM optimizer [1]} for 10,000 iterations with default parameters. We use a batch size of 10 samples to evaluate both the \(MSE_{\rm data}\)  and \(MSE_{\rm PDE}\)  term. For comparison, we also train the same neural network by removing the physics, that is not considering the partial differential equation loss shown equation REF . Additionally, we train a regular physics-informed neural network, where we use the Cartesian coordinates as the input of the neural network. We implemented the Eikonal equation in the \(MSE_{\rm PDE}\)  computing \(\nabla u\)  through automatic differentiation. This network also had only single hidden layer of 100 neurons. We also tested deeper architectures without any meaningful change in the results, using the same optimization procedure as described above.
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